<h2>Problem 207</h2>
<div style="color:#666;font-size:80%;">06 September 2008</div><br />
<div class="problem_content">
<p>For some positive integers <var>k</var>, there exists an integer partition of the form&nbsp;&nbsp; 4<img src="" style="display:none;" alt="^(" /><sup>t</sup><img src="" style="display:none;" alt=")" /> = 2<img src="" style="display:none;" alt="^(" /><sup>t</sup><img src="" style="display:none;" alt=")" /> + <var>k</var>,<br />
where 4<img src="" style="display:none;" alt="^(" /><sup>t</sup><img src="" style="display:none;" alt=")" />, 2<img src="" style="display:none;" alt="^(" /><sup>t</sup><img src="" style="display:none;" alt=")" />, and <var>k</var> are all positive integers and <var>t</var> is a real number.</p>

<p>The first two such partitions are 4<img src="" style="display:none;" alt="^(" /><sup>1</sup><img src="" style="display:none;" alt=")" /> = 2<img src="" style="display:none;" alt="^(" /><sup>1</sup><img src="" style="display:none;" alt=")" /> + 2 and 4<img src="" style="display:none;" alt="^(" /><sup>1.5849625...</sup><img src="" style="display:none;" alt=")" /> = 2<img src="" style="display:none;" alt="^(" /><sup>1.5849625...</sup><img src="" style="display:none;" alt=")" /> + 6.</p>

<p>Partitions where <var>t</var> is also an integer are called <i>perfect</i>.<br /> 
For any <var>m</var> <img src='images/symbol_ge.gif' width='10' height='12' alt='&ge;' border='0' style='vertical-align:middle;' /> 1 let P(<var>m</var>) be the proportion of such partitions that are perfect with <var>k</var> <img src='images/symbol_le.gif' width='10' height='12' alt='&le;' border='0' style='vertical-align:middle;' /> <var>m</var>.<br />
Thus P(6) = 1/2.</p>

<p>In the following table are listed some values of P(<var>m</var>)</p>
<p>&nbsp;&nbsp;&nbsp;P(5) = 1/1<br />
&nbsp;&nbsp;&nbsp;P(10) = 1/2<br />
&nbsp;&nbsp;&nbsp;P(15) = 2/3<br />
&nbsp;&nbsp;&nbsp;P(20) = 1/2<br />
&nbsp;&nbsp;&nbsp;P(25) = 1/2<br />
&nbsp;&nbsp;&nbsp;P(30) = 2/5<br />
&nbsp;&nbsp;&nbsp;...<br />
&nbsp;&nbsp;&nbsp;P(180) = 1/4<br />
&nbsp;&nbsp;&nbsp;P(185) = 3/13</p>


<p>Find the smallest <var>m</var> for which P(<var>m</var>) <img src='images/symbol_lt.gif' width='10' height='10' alt='&lt;' border='0' style='vertical-align:middle;' /> 1/12345</p>
</div><br />
